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On Distributions of Ratios

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  • Simon A. Broda
  • Raymond Kan

    (University of Toronto)

Abstract

A large number of exact inferential procedures in statistics and econometrics involve the sampling distribution of ratios of random variables. If the denominator variable is positive, then tail probabilities of the ratio can be expressed as those of a suitably defined difference of random variables. If in addition, the joint characteristic function of numerator and denominator is known, then standard Fourier inversion techniques can be used to reconstruct the distribution function from it. Most research in this field has been based on this correspondence, but which breaks down when both numerator and denominator are supported on the entire real line. The present manuscript derives inversion formulae and saddlepoint approximations that remain valid in this case, and reduce to known results when the denominator is almost surely positive. Applications include the IV estimator of a structural parameter in a just identified equation.

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Bibliographic Info

Paper provided by Universiteit van Amsterdam, Dept. of Econometrics in its series UvA-Econometrics Working Papers with number 13-10.

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Date of creation: 22 Dec 2013
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Handle: RePEc:ame:wpaper:1310

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Postal: Dept. of Econometrics, Universiteit van Amsterdam, Valckenierstraat 65, NL - 1018 XE Amsterdam, The Netherlands
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Web page: http://www.ase.uva.nl/uva-econometrics
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  1. Maddala, G S & Jeong, Jinook, 1992. "On the Exact Small Sample Distribution of the Instrumental Variable Estimator," Econometrica, Econometric Society, vol. 60(1), pages 181-83, January.
  2. Peter C. B. Phillips, 2005. "A Remark on Bimodality and Weak Instrumentation in Structural Equation Estimation," Cowles Foundation Discussion Papers 1540, Cowles Foundation for Research in Economics, Yale University.
  3. Russell Davidson & James G. MacKinnon, 2006. "Bootstrap Inference in a Linear Equation Estimated by Instrumental Variables," Working Papers 1024, Queen's University, Department of Economics.
  4. Giovanni Forchini, 2005. "On the Bimodality of the Exact Distribution of the TSLS Estimator," Monash Econometrics and Business Statistics Working Papers 14/05, Monash University, Department of Econometrics and Business Statistics.
  5. Charles R. Nelson & Richard Startz, 1988. "Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator," NBER Technical Working Papers 0068, National Bureau of Economic Research, Inc.
  6. Simon A. Broda, 2013. "Tail probabilities and partial moments for quadratic forms in multivariate generalized hyperbolic random vectors," UvA-Econometrics Working Papers 13-04, Universiteit van Amsterdam, Dept. of Econometrics.
  7. Anderson, T W & Sawa, Takamitsu, 1973. "Distributions of Estimates of Coefficients of a Single Equation in a Simultaneous System and Their Asymptotic Expansions," Econometrica, Econometric Society, vol. 41(4), pages 683-714, July.
  8. Russell Davidson & James Mackinnon, 2009. "Wild bootstrap tests for IV regression," Working Papers halshs-00443550, HAL.
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  11. Alfonso Flores-Lagunes, 2007. "Finite sample evidence of IV estimators under weak instruments," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 22(3), pages 677-694.
  12. Soujin Wang, 1990. "Saddlepoint approximations in resampling analysis," Annals of the Institute of Statistical Mathematics, Springer, vol. 42(1), pages 115-131, March.
  13. Hillier, Grant, 2006. "Yet More On The Exact Properties Of Iv Estimators," Econometric Theory, Cambridge University Press, vol. 22(05), pages 913-931, October.
  14. Woglom, Geoffrey, 2001. "More Results on the Exact Small Sample Properties of the Instrumental Variable Estimator," Econometrica, Econometric Society, vol. 69(5), pages 1381-89, September.
  15. Holly, A & Phillips, P C B, 1979. "A Saddlepoint Approximation to the Distribution of the k-Class Estimator of a Coefficient in a Simultaneous System," Econometrica, Econometric Society, vol. 47(6), pages 1527-47, November.
  16. Shephard, N.G., 1991. "From Characteristic Function to Distribution Function: A Simple Framework for the Theory," Econometric Theory, Cambridge University Press, vol. 7(04), pages 519-529, December.
  17. Knight, John L., 1986. "Non-Normal Errors and the Distribution of OLS and 2SLS Structural Estimators," Econometric Theory, Cambridge University Press, vol. 2(01), pages 75-106, April.
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